Chapter 12 The Black-Scholes Formula

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Chapter 12Chapter 12The Black-Scholes FormulaThe Black-Scholes Formula

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Black-Scholes Formula

Call Options:

Put Options:

where

anddln S / K r T

T1

212

( ) ( )

d d T2 1

C S,K, ,r,T, Se N d Ke N d- T -rT( ) = ( ) ( ) 1 2

P S,K, ,r,T, Ke N d Se N d-rT - T ( ) = ( ) ( ) 2 1

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Black-Scholes (BS) assumptions

Assumptions about stock return distribution Continuously compounded returns on the stock are

normally distributed and independent over time (no “jumps”)

The volatility of continuously compounded returns is known and constant

Future dividends are known, either as dollar amount or as a fixed dividend yield

Assumptions about the economic environment The risk-free rate is known and constant There are no transaction costs or taxes It is possible to short-sell costlessly and to borrow at the

risk-free rate

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Applying BS to other assets

Call Options:

where

,

and

d

ln F S / F K T

T

TP

TP

1

0 021

2, ,( ) ( )

d d T2 1

C F S ,F K , ,T F S N d F K N dTP

TP

TP

TP( ) = ( ) ( )0 0 0 1 0 2, , , ,( ) ( ) ( ) ( )

F K KeTP -rT

0, ( ) =

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Applying BS to other assets (cont.)

Options on stocks with discrete dividends:

The prepaid forward price for stock with discrete dividends is:

Examples 12.3 and 12.1: S = $41, K = $40, = 0.3, r = 8%, t = 0.25, Div = $3 in one

month PV (Div) = $3e-0.08/12 = $2.98 Use $41– $2.98 = $38.02 as the stock price in BS formula The BS European call price is $1.763 Compare this to European call on stock without dividends:

$3.399

F S S PV DivTP

T0 0 0, ,( ) = ( )

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Applying BS to other assets (cont.)

Options on currencies: The prepaid forward price for the currency is:

where x is domestic spot rate and rf is foreign interest rate

Example 12.4: x = $0.92/ , K = $0.9, = 0.10, r = 6%, T = 1, and

= 3.2% The dollar-denominated euro call price is $0.0606 The dollar-denominated euro put price is $0.0172

F x x eTP -r Tf

0 0, ( ) =

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Applying BS to other assets

Options on futures: The prepaid forward price for a futures contract is the PV of the

futures price. Therefore:

where

and

Example 12.5: Suppose 1-yr. futures price for natural gas is $2.10/MMBtu, r =

5.5% Therefore, F=$2.10, K=$2.10, and = 5.5% If = 0.25, T= 1, call price = put price = $0.197721

C F,K, ,r,t Fe N d Ke N d-rt -rt( ) = ( ) ( ) 1 2

d

ln F / K T

T1

212

d d T2 1

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Option Greeks

What happens to option price when one input changes? Delta (): change in option price when stock price increases by

$1 Gamma (): change in delta when option price increases by $1 Vega: change in option price when volatility increases by 1% Theta (): change in option price when time to maturity

decreases by 1 day Rho (): change in option price when interest rate increases by

1% Greek measures for portfolios

The Greek measure of a portfolio is weighted average of Greeks

of individual portfolio components:

portfolio i i

i

n

1

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Option Greeks (cont.)

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Option elasticity (: describes the risk of the option relative to the risk of the stock in percentage terms: If stock

price (S) changes by 1%, what is the percent change in the value of the option (C)?

Example 12.8: S = $41, K = $40, = 0.30, r = 0.08, T = 1, = 0 Elasticity for call: = S /C = $41 x 0.6911 / $6.961 = 4.071 Elasticity for put: = S /C = $41 x – 0.3089 / $2.886 = – 4.389

%

%

C

S

CC

SS

S C

S S

S

C

change in

change in

change in

change in

change in

change in

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Option elasticity (: (cont.)

The volatility of an option:

The risk premium of an option:

The Sharpe ratio of an option:

where | . | is the absolute value, : required return on option, : expected return on stock, and r: risk-free rate

option stock

r r( )

Sharpe ratio for call Sharpe ratio for stock

r

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Profit diagrams before maturity

For purchased call option:

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Implied volatility

The volatility of the returns consistent with observed option prices and the pricing model (typically Black-Scholes) One can use the implied volatility from an option with an

observable price to calculate the price of another option on the same underlying asset

Checking the uniformity of implied volatilities across various options on the same underlying assets allows one to verify the validity of the pricing model in pricing those options

In practice implied volatilities of in, at, and out-of-the money options are generally different resulting in the volatility skew

Implied volatilities of puts and calls with same strike and time to expiration must be the same if options are European because of put-call parity

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Implied volatility (cont.)

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Perpetual American options

Perpetual American options (options that never expire) are optimally exercised when the underlying asset ever reaches the optimal exercise barrier H* (if = 0, H* = infinity) For a perpetual call option: and

For a perpetual put option: and

where and

H Kh

h*

1

1 1Pr ice

K

h

h

h

S

K

h

1

1

11

1 1

hr r r

1 2 2

2

2

1

2

1

2

2

H Kh

h*

2

2 1Pr ice

K

h

h

h

S

K

h

1

1

2

2

2

2

hr r r

2 2 2

2

2

1

2

1

2

2

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Chapter 12 The Black-Scholes Formula